Precalculus Examples

$2 \log (x) - \log (7) = \log (567)$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$2 \log (x) - \log (7) - \log (567) = 0$

Step 2

Simplify the left side.

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Step 2.1

Simplify $2 \log (x) - \log (7) - \log (567)$ .

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Step 2.1.1

Simplify $2 \log (x)$ by moving $2$ inside the logarithm.

$\log (x^{2}) - \log (7) - \log (567) = 0$

Step 2.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{x^{2}}{7}) - \log (567) = 0$

Step 2.1.3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{\frac{x^{2}}{7}}{567}) = 0$

Step 2.1.4

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{x^{2}}{7} \cdot \frac{1}{567}) = 0$

Step 2.1.5

Combine.

$\log (\frac{x^{2} \cdot 1}{7 \cdot 567}) = 0$

Step 2.1.6

Multiply.

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Step 2.1.6.1

Multiply $x^{2}$ by $1$ .

$\log (\frac{x^{2}}{7 \cdot 567}) = 0$

Step 2.1.6.2

Multiply $7$ by $567$ .

$\log (\frac{x^{2}}{3969}) = 0$

Step 3

Rewrite $\log (\frac{x^{2}}{3969}) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$10^{0} = \frac{x^{2}}{3969}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $\frac{x^{2}}{3969} = 10^{0}$ .

$\frac{x^{2}}{3969} = 10^{0}$

Step 4.2

Multiply both sides of the equation by $3969$ .

$3969 \frac{x^{2}}{3969} = 3969 \cdot 10^{0}$

Step 4.3

Simplify both sides of the equation.

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Step 4.3.1

Simplify the left side.

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Step 4.3.1.1

Cancel the common factor of $3969$ .

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Step 4.3.1.1.1

Cancel the common factor.

$3969 \frac{x^{2}}{3969} = 3969 \cdot 10^{0}$

Step 4.3.1.1.2

Rewrite the expression.

$x^{2} = 3969 \cdot 10^{0}$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Move the decimal point in $3969$ to the left by $3$ places and increase the power of $10^{0}$ by $3$ .

$x^{2} = 3.969 \cdot 10^{3}$

Step 4.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{3.969 \cdot 10^{3}}$

Step 4.5

Simplify $\pm \sqrt{3.969 \cdot 10^{3}}$ .

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Step 4.5.1

Rewrite $3.969 \cdot 10^{3}$ as $0.3969 \cdot 10^{4}$ .

$x = \pm \sqrt{0.3969 \cdot 10^{4}}$

Step 4.5.2

Rewrite $\sqrt{0.3969 \cdot 10^{4}}$ as $\sqrt{0.3969} \cdot \sqrt{10^{4}}$ .

$x = \pm \sqrt{0.3969} \cdot \sqrt{10^{4}}$

Step 4.5.3

Evaluate the root.

$x = \pm 0.63 \cdot \sqrt{10^{4}}$

Step 4.5.4

Rewrite $10^{4}$ as ${(10^{2})}^{2}$ .

$x = \pm 0.63 \cdot \sqrt{{(10^{2})}^{2}}$

Step 4.5.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0.63 \cdot 10^{2}$

Step 4.5.6

Move the decimal point in $0.63$ to the right by $1$ place and decrease the power of $10^{2}$ by $1$ .

$x = \pm 6.3 \cdot 10^{1}$

Step 4.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 6.3 \cdot 10^{1}$

Step 4.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 6.3 \cdot 10^{1}$

Step 4.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 6.3 \cdot 10^{1}, - 6.3 \cdot 10^{1}$

Step 5

Exclude the solutions that do not make $2 \log (x) - \log (7) = \log (567)$ true.

$x = 6.3 \cdot 10^{1}$

Step 6

The result can be shown in multiple forms.

Scientific Notation:

$x = 6.3 \cdot 10^{1}$

Expanded Form:

$x = 63$